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Part V Appendixes
Appendix A Conversion Factors
The following table expresses the definitions of miscellaneous units of measure as exact numerical multiples of coherent SI units, and provides multiplying factors for converting numbers and miscellaneous units to corresponding new numbers and SI units.
The digits of each numerical entry following E represent a power of 10. An asterisk preceding each number expresses an exact definition. For example, the entry "*2.54E-2" expresses the fact that 1 inch=2.54x 10- 2
meter, exactly, by definition. Most of the definitions are extracted from National Bureau of Standards documents. Numbers not preceded by an asterisk are only approximate representations of definitions, or are the results of physical measurements.
This appendix was abstracted from The International Systems of Units-Physical Constants and Conversion Factors. E.A. Mechtly, Second Revision. NASA SP-7012, Washington, D.C. (1973). Permission to use this material was obtained from the Scientific and Technical Information Office, NASA, Washington, D.C.
Table A.l
To convert from: to: multiply by:
atmosphere newton/meter2 *1.013E5 bar newton/meter2 *l.OOES British thermal unit (mean) joule 1.05587E3 British thermal unit joule 1.054350E3
(thermochemical) British thermal unit (39°F) joule 1.05967E3 British thermal unit ( 60° F) joule 1.05468E3 calorie (International Steam Table) joule 4.1868 calorie (mean) joule 4.19002 calorie (thermochemical) joule *4.184 calorie (15oC) joule 4.18580 calorie (20° C) joule 4.18190 calorie (kilogram, International joule 4.1868E3
Steam Table) continued overleaf
299
300
To convert from:
calorie (kilogram, mean) calorie (kilogram, thermochemical) Celsius (temperature) centimeter of mercury (Oo C) centimeter of water (4° C) electron volt erg Fahrenheit (temperature) Fahrenheit (temperature) fluid ounce (U.S.) foot foot of water (39.2° F) gallon (U.S. dry) gallon (U.S. liquid) horsepower (550 foot lbf/secondJ inch kilocalorie (International Steam
Table) kilocalorie (mean) kilocalorie (thermochemical) kilogram mass kilogram force (kgf) lbf (pound force, avoirdupois) Ibm (pound mass, avoirdupois) liter micron mile (U.S. statute) pascal poise pound force (lbf avoirdupois) pound mass (Ibm avoirdupois) quart (U.S. dry) quart (U.S. liquid) Rankine (temperature) slug ton (long) ton (metric) ton (short, 2000 Jb) Torr (OoC)
Table A.l (Continued)
to:
joule joule kelvin newton/meter2
newton(meter 2
joule joule kelvin Celsius meter 3
meter newton/meter2
meter 3
meter3
watt meter joule
joule joule kilogram newton newton kilogram meter 3
meter meter newton(meter2
newton second(meter2
newton kilogram meter 3
meter 3
kelvin kilogram kilogram kilogram kilogram newton(meter2
Appendix A
multiply by:
4.19002E3 *4.184E3
tK =tc +273.15 1.33322E3 9.80638E1 1.6021917E-19
*l.OOE-7 fK = (5/9)(1F +459.67) tc=(5/9)(tF-32)
*2.957 35295625E5 *3.048E-1 2.98898E3
*4.404 883 770 86E-3 *3.785411784E-3
7.4569987E2 *2.54E-2 4.1868E3
4.19002E3 4.184E3
*1.00 *9.80665 *4.448 221615 260 5 *4.535 923 7E-1 *l.OOE-3 *l.OOE-6 *1.609 344E3 *1.00 *l.OOE-1 *4.4482216152605 *4.535 923 7E-1 *1.101220942 715E-3 9.4635925E-4 (K = (5(9)t R
1.459 39029EI *1.016046908 8E3 *l.OOE3 *9.0718474E2
1.33322E2
Appendix B
Physical Parameters for Prediction of Transport Coefficients
Contents
Table B.l. Intermolecular Force Parameters and Critical Properties, p. 301 Table B.2. Leonard-Jones Patentials as Determined from Viscosity Data,
p. 303 Table B.3. Stockmayer-Potential Parameters, p. 305 Table B.4. Values of the Collision Integral Q v for Viscosity Based on the
Leonard-Jones Potential, p. 306 Table B.S. Values of the Collision Integral !lv Based on the Leonard
Jones Potential, p. 307 Table B.6. Collision Integrals Q. for Viscosity as Calculated by Use of
the Stockmayer Potential, p. 308 Notation for the Tables, p. 309 References for the Tables, p. 309
Table B.l. Intermolecular Force Parameters and Critical Properties"
Lennard-Jones parametersb Critical constants'
Molecular weight (5 e/K I; p, V, X 103
Substance M (A) (K) (K) (atm) (m 3 kmol- 1)
Light elements H2 2.016 2.915 38.0 33.3 12.80 65.0 He 4.003 2.576 10.2 5.26 2.26 57.8
continued overleaf"
301
302 Appendix B
Table B.l (Continued)
Lennard-Janes parameters• Critical constants'
Molecular ------ ---~---·--
weight (J e/K I; p, V, X 103
Substance M (Al (K) (K) (atm) (m3 kmol- 1)
Noble gases Ne 20.183 2.789 35.7 44.5 26.9 41.7
Ar 39.944 3.418 124. 151. 48.0 75.2
Kr 83.80 3.498 225. 209.4 54.3 92.2
Xe 131.3 4.055 229. 289.8 58.0 118.8
Simple po/yatomic substances Air 28.97d 3.617 97.0 132.d 36.4d 86.6d
Nz 28.02 3.681 91.5 126.2 33.5 90.1
02 32.00 3.433 113. 154.4 49.7 74.4
03 48.00 268. 67. 89.4
co 28.01 3.590 110. 133. 34.5 93.1
col 44.01 3.996 190. 304.2 72.9 94.0
NO 30.01 3.470 119. 180. 64. 57.
N20 44.02 3.879 220. 309.7 71.7 96.3
S02 64.07 4.290 252. 430.7 77.8 122.
Fz 38.00 3.653 112.
Cl 2 70.91 4.115 357. 417. 76.1 124.
Br2 159.83 4.268 520. 584. 102. 144.
12 253.82 4.982 550. 800.
Hydrocarbons CH4 16.04 3.822 137. 190.7 45.8 99.3
CzHz 26.04 4.221 185. 309.5 61.6 113.
CzH• 28.05 4.232 205. 282.4 50.0 124.
CzH6 30.07 4.418 230. 305.4 48.2 148.
C3H6 42.08 365.0 45.5 181.
C3H 8 44.09 5.061 254. 370.0 42.0 200.
n-C.H,o 58.12 425.2 37.5 255.
i-C4 H, 0 58.12 5.341 313. 408.1 36.0 263.
n-C5H,z 72.15 5.769 345. 469.8 33.3 311.
IJ-C6H14 86.17 5.909 413. 507.9 29.9 368.
n-C7H, 6 100.20 540.2 27.0 426.
n-C 8 H, 8 114.22 7.451 320. 569.4 24.6 485.
n-C9 Hzo 128.25 595.0 22.5 543. Cyclohexane 84.16 6.093 324. 553. 40.0 308.
C6 H 6 78.11 5.270 440. 562.6 48.6 260. contmued
Appendix B
Substance
CH4
CH 3Cl CH2Cl 2
CHC13
CC1 4
C 2 N 2
cos CS 2
Molecular weight
M
16.04 50.49 84.94
119.39 153.84
52.04 60.08 76.14
Table B.l (Continued)
Lennard-Jones parameters•
(J B/K 7;, (AJ (K) (K) -----~
Other organic compounds 3.822 137. 190.7 3.375 855. 416.3 4.759 406. 510.0 5.430 327. 536.6 5.881 327. 556.4 4.38 339. 400. 4.13 335. 378. 4.438 488. 552.
"Table used with permisswn from Bird et a1' 11
303
Critical constants'
p, J!; X 103
(atm) (m 3 kmol- 1)
45.8 99.3 65.9 143. 60. 54. 240. 45.0 276. 59. 61. 78. 170.
•values of a and cjK are from J. 0. Hirschfelder. C. F. Curtiss. and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954). The above values are computed from viscosity data and are applicable for temperatures above 100 K.
··values of 7;, p, and v; are from K. A. Kobe and R. E. Lynn, Jr., Chern. Rev. 52, 117-236 (1952); and American Petroleum Institute Research Project, Volume 44, (F. D. Rossini, ed.), Carnegie Institute of Technology (1952).
'For air, the molecular weight M and the pseudocritical properties 7;, p" and V,. have been calculated from the average composition of dry air, as given in International Critical Tables, Volume I, p. 393 (1926).
Table B.2. Lennard-Janes Potentials as Determined from Viscosity Dataa
b0 x 103
a(A) Molecule Compound (m 3 kmol- 1) B/k (K)
A Argon 46.08 3.542 93.3 He Helium 20.95 2.551' 10.22 Kr Krypton 61.62 3.655 178.9 Ne Neon 28.30 2.820 32.8 Xe Xenon 83.66 4.047 231.0 Air Air 64.50 3.711 78.6 AsH 3 Arsine 89.88 4.145 259.8 BC13 Boron chloride 170.1 5.127 337.7 BF3 Boron fluoride 93.35 4.198 186.3 B(OCH 3h Methyl borate 210.3 5.503 396.7 Br2 Bromine 100.1 4.296 507.9 CC14 Carbon tetrachloride 265.5 5.947 322.7 CF4 Carbon tetrafluoride 127.9 4.662 134.0 CHC13 Chloroform 197.5 5.389 340.2 CH2 Cl 2 Methylene chloride 148.3 4.898 356.3 CH3 Br Methyl bromide 88.14 4.118 449.2 CH 3Cl Methyl chloride 92.31 4.182 350 CH30H Methanol 60.17 3.626 481.8 CH4 Methane 66.98 3.758 148.6 co Carbon monoxide 63.41 3.690 91.7
continued overleaf
304 Appendix B
Table B.2 (Continued)
b0 x 103
Molecule Compound (m 3 kmol- 1 ) a(A) ejk (K)
cos Carbonyl sulfide 88.91 4.130 336.0 C02 Carbon dioxide 77.25 3.941 195.2 CS 2 Carbon disulfide 113.7 4.483 467
CzHz Acetylene 82.79 4.033 231.8
CzH4 Ethylene 91.06 4.163 224.7
CzH6 Ethane 110.7 4.443 215.7 C2H5Cl Ethyl chloride 148.3 4.898 300 C2H 5 0H Ethanol 117.3 4.530 362.6
CzNz Cyanogen 104.7 4.361 348.6 CH 30CH 3 Methyl ether 100.9 4.307 395.0
CH 2CHCH3 Propylene 129.2 4.678 298.9 CH3CCH Methyl acetylene 136.2 4.761 251.8
C3H6 Cyclopropane 140.2 4.807 248.9
C3Hs Propane 169.2 5.118 237.1 n-C3 H 70H n-Propyl alcohol 118.8 4.549 576.7 CH 3COCH3 Acetone 122.8 4.600 560.2 CH 3COOCH 3 Methyl acetate 151.8 4.936 469.8
n-C4 H 1o n-Butane 130.0 4.687 531.4 iso-C4H 10 !so butane 185.6 5.278 330.1
C 2H 50CzHs Ethyl ether 231.0 5.678 313.8 CH 3COOC2H 5 Ethyl acetate 178.0 5.205 521.3
n-C 5H 1z n-Pentane 244.2 5.784 341.1 C(CH3)4 2,2-Dimethyl propane 340.9 6.464 193.4
C6H6 Benzene 193.2 5.349 412.3
C6H12 Cyclohexane 298.2 6.182 297.1 n-C6H 14 n-Hexane 265.7 5.949 399.3
Cl 2 Chlorine 94.65 4.217 316.0
Fz Fluorine 47.75 3.357 112.6 HBr Hydrogen bromide 47.58 3.353 449 HCN Hydrogen cyanide 60.37 3.630 569.1 HCl Hydrogen chloride 46.98 3.339 344.7 HF Hydrogen fluoride 39.37 3.148 330 HI Hydrogen iodide 94.24 4.211 288.7
Hz Hydrogen 28.51 2.827 59.7 H 20 Water 23.25 2.641 809.1
HzOz Hydrogen peroxide 93.24 4.196 289.3 H 2S Hydrogen sulfide 60.02 3.623 301.1 Hg Mercury 33.03 2.969 750 HgBr2 Mercuric bromide 165.5 5.080 686.2 HgC1 2 Mercuric chloride 118.9 4.550 750 Hgl 2 Mercuric iodide 224.6 5.625 695.6
lz Iodine 173.4 5.160 474.2
NH 3 Ammonia 30.78 2.900 558.3 NO Nitric oxide 53.74 3.492 116.7 NOCl Nitrosyl chloride 87.75 4.112 395.3
continued
Appendix B
Molecule
Nz N 20 02 PH 3
SF6
soz SiF4
SiH4
SnBr4
UF6
Table B.2 (Continued)
b0 x 103
Compound (m 3 kmol-I)
Nitrogen 69.14 Nitrous oxide 70.80 Oxygen 52.60 Phosphine 79.63 Sulfur hexafluoride 170.2 Sulfur dioxide 87.75 Silicon tetrafluoride 146.7 Silicon hydride 85.97 Stannic bromide 329.0 Uranium hexafluoride 268.1
u(A)
3.798 3.828 3.467 3.981 5.128 4.112 4.880 4.084 6.388 5.967
305
o/k (K)
71.4 232.4 106.7 251.5 222.1 335.4 171.9 207.6 563.7 236.8
"R. A. Svehla. NASA Techmcal Report R-132, Lewis Research Center, Cleveland, Ohio (1962); table used with permissiOn from Reid and Sherwood.' 21
'b 0 =~nN0a3 , where N 0 is Avogadro's number. 'The potential a was determined by quantum mechanical formulas.
Table B.J. Stockmayer-Potential Parameters"
Dipole moment <T o/k b0 x 103
Molecule 11 (debyes) (A) (K) (m 3 kmoJ-I) bmax
H20 1.85 2.52 775 20.2 1.0 NH 3 1.47 3.15 358 39.5 0.7 HCl 1.08 3.36 328 47.8 0.34 1-lBr 0.80 3.41 417 50.0 0.14 HI 0.42 4.13 313 88.9 0.029
soz 1.63 4.04 347 83.2 0.42 H 2S 0.92 3.49 343 53.6 0.21 NOCI 1.83 3.53 690 55.5 0.4 CHC1 3 1.013 5.31 355 189 0.07 CH 2Cl2 1.57 4.52 483 117 0.2 CH 3Cl 1.87 3.94 414 77.2 0.5 CH 3 Br 1.80 4.25 382 96.9 0.4 C2H 5Cl 2.03 4.45 423 Ill 0.4 CH 30H 1.70 3.69 417 63.4 0.5 C2H 50H 1.69 4.31 431 101 0.3 n-C3 H 70H 1.69 4.71 495 132 0.2 i-C3 H 70H 1.69 4.64 518 126 0.2 (CH 3 lz0 1.30 4.21 432 94.2 0.19 (CzHslzO 1.15 5.49 362 209 0.08 (CH 3)zCOb 1.20 3.82 428 70.1 1.3
contmued overleaf
306 Appendix B
Table B.J (Continued)
Dipole movement () &/k b0 x 103
Molecule J1 (debyes) (Al (K) (m 3 kmo1- 1 ) <}max
CH 3COOCH 3 1.72 5.04 418 162 0.2 CH 3COOC2H 5 1.78 5.24 499 182 0.16 CH 3NO/ 2.15 4.16 290 90.8 2.3
"L. Monchick and E. A. Mason. J. Chern. Phys. 35. 1676 (1961); table used with permission from Reid and Sherwood.<2> "From G. A. Bottomley and T. H. Spurling. Austral. J. Chern. 16. I (1963). Monchick and Mason show that a=4.50 A; r.fk=549 K.
Table B.4. Values of the Collision Integral O..Jor Viscosity Based on the Lennard-Janes Potential"
T*=kT/Eb Qvb kT/£ n,. kT/c Qv
------··~-·---~·------- .. ---- --------
0.30 2. 785 1.65 1.264 4.0 0.9700 0.35 2.628 1.70 1.248 4.1 0.9649 0.40 2.492 1.75 1.234 4.2 0.9600 0.45 2.368 1.80 1.221 4.3 0.9553 0.50 2.257 1.85 1.209 4.4 0.9507 0.55 2.156 1.90 1.197 4.5 0.9464 0.60 2.065 1.95 1.186 4.6 0.9422 0.65 1.982 2.00 1.175 4.7 0.9382 0.70 1.908 2.1 1.156 4.8 0.9343 0.75 1.841 2.2 1.138 4.9 0.9305 0.80 1.780 2.3 1.122 5.0 0.9269 0.85 1.725 2.4 1.107 6.0 0.8963 0.90 1.675 2.5 1.093 7.0 0.8727 0.95 1.629 2.6 1.081 8.0 0.8538 1.00 1.587 2.7 1.069 9.0 0.8379 1.05 1.549 2.8 1.058 10 0.8242 1.10 1.514 2.9 1.048 20 0.7432 1.15 1.482 3.0 1.039 30 0.7005 1.20 1.452 3.1 1.030 40 0.6718 1.25 1.424 3.2 1.022 50 0.6504 1.30 1.399 3.3 1.014 60 0.6335 1.35 1.375 3.4 1.007 70 0.6194 1.40 1.353 3.5 0.9999 80 0.6076 1.45 1.333 3.6 0.9932 90 0.5973
'untmued
Appendix B
1.50 1.55 1.60
Table B.4 (Continued)
1.314 1.296 1.279
kT/e
3.7 3.8 3.9
0.9870 0.9811 0.9755
kT/e
100 200 400
0.5882 0.5320 0.4811
"Table used with permission from Reid and Sherwood.<" •Hirschfelder, Curtiss, and Bird13) use the symbol T* for kT /E and Q<2.2J*
for Q". Bromley and Wilke14l use (kT/e) 112 V/W2(2) forf1(kT/e). More complete tables of these functions are given In the two references cited.
Table B.5. Values ~f the Collision Integral Ov Based on the Lennard-Janes Potential"
kT/eb nvb kT/e nv kT/r. nv
0.30 2.662 1.65 1.153 4.0 0.8836 0.35 2.476 1.70 1.140 4.1 0.8788 0.40 2.318 1.75 1.128 4.2 0.8740 0.45 2.184 1.80 1.116 4.3 0.8694 0.50 2.066 1.85 1.105 4.4 0.8652 0.55 1.966 1.90 1.094 4.5 0.8610 0.60 1.877 1.95 1.084 4.6 0.8568 0.65 1.798 2.00 1.075 4.7 0.8530 0.70 1.729 2.1 1.057 4.8 0.8492 0.75 1.667 2.2 1.041 4.9 0.8456 0.80 1.612 2.3 1.026 5.0 0.8422 0.85 1.562 2.4 1.012 6 0.8124 0.90 1.517 2.5 0.9996 7 0.7896 0.95 1.476 2.6 0.9878 8 0.7712 1.00 1.439 2.7 0.9770 9 0.7556 1.05 1.406 2.8 0.9672 10 0.7424 1.10 1.375 2.9 0.9576 20 0.6640 1.15 1.346 3.0 0.9490 30 0.6232 1.20 1.320 3.1 0.9406 40 0.5960 1.25 1.296 3.2 0.9328 50 0.5756 1.30 1.273 3.3 0.9256 60 0.5596 1.35 1.253 3.4 0.9186 70 0.5464 1.40 1.233 3.5 0.9120 80 0.5352 1.45 1.215 3.6 0.9058 90 0.5256 1.50 1.198 3.7 0.8998 100 0.5130 1.55 1.182 3.8 0.8942 200 0.4644 1.60 1.167 3.9 0.8888 400 0.4170
"From J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc. New York (1954); table used with permission of Reid and Sherwood.'"
'Hirschfelder used the symbols T* for kT/e and 0°·"* in place of 0 0 .
307
308 Appendix B
Table B.6. Collision Integrals D.vfor Viscosity as Calculated by Use of the Stockmayer Potential"·b
~ 0' 0.25 0.50 0.75 1.0 1.5 2.0 2.5
0.1 4.1005 4.266 4.833 5.742 6.729 8.624 10.34 11.89 0.2 3.2626 3.305 3.516 3.914 4.433 5.570 6.637 7.618 0.3 2.8399 2.836 2.936 3.168 3.511 4.329 5.126 5.874 0.4 2.5310 2.522 2.586 2.749 3.004 3.640 4.282 4.895 0.5 2.2837 2.277 2.329 2.460 2.665 3.187 3.727 4.249 0.6 2.0838 2.081 2.130 2.243 2.417 2.862 3.329 3.786 0.7 1.9220 1.924 1.970 2.072 2.225 2.614 3.028 3.435 0.8 1.7902 1.795 1.840 1.934 2.070 2.417 2.788 3.156 0.9 1.6823 1.689 1.733 1.820 1.944 2.258 2.596 2.933 1.0 1.5929 1.601 1.644 1.725 1.838 2.124 2.435 2.746 1.2 1.4551 1.465 1.504 1.574 1.670 1.913 2.181 2.451 1.4 1.3551 1.365 1.400 1.461 1.544 1.754 1.989 2.228 1.6 1.2800 1.289 1.321 1.374 1.447 1.630 1.838 2.053 1.8 1.2219 1.231 1.259 1.306 1.370 1.532 1.718 1.912 2.0 1.1757 1.184 1.209 1.251 1.307 1.451 1.618 1.795 2.5 1.0933 1.100 1.119 1.150 1.193 1.304 1.435 1.578 3.0 1.0388 1.044 1.059 1.083 1.117 1.204 1.310 1.428 3.5 0.99963 1.004 1.016 1.035 1.062 1.133 1.220 1.319 4.0 0.96988 0.9732 0.9830 0.9991 1.021 1.079 1.153 1.236 5.0 0.92676 0.9291 0.9360 0.9473 0.9628 1.005 1.058 1.121 6.0 0.98616 0.8979 0.9030 0.9114 0.9230 0.9545 0.9955 1.044 7.0 0.87272 0.8741 0.8780 0.8845 0.8935 0.9181 0.9505 0.9893 8.0 0.85379 0.8549 0.8580 0.8632 0.8703 0.8901 0.9164 0.9482 9.0 0.83795 0.8388 0.8414 0.8456 0.8515 0.8678 0.8895 0.9160
10.0 0.82435 0.8251 0.8273 0.8308 0.8356 0.8493 0.8676 0.8901 12.0 0.80184 0.8024 0.8039 0.8065 0.8101 0.8201 0.8337 0.8504 14.0 0.78363 0.7840 0.7852 0.7872 0.7899 0.7976 0.8081 0.8212 16.0 0.76834 0.7687 0.7696 0.7712 0.7733 0.7794 0.7878 0.7983 18,0 0.75518 0.7554 0.7562 0.7575 0.7592 0.7642 0.7711 0.7797 20.0 0.74364 0.7438 0.7445 0.7455 0.7470 0.7512 0.7569 0.7642 25.0 0.71982 0.7200 0.7204 0.7211 0.7221 0.7250 0.7289 0.7339 30.0 0.70097 0.7011 0.7014 0.7019 0.7026 0.7047 0.7076 0.7112 35.0 0.68545 0.6855 0.6858 0.6861 0.6867 0.6883 0.6905 0.6932 40.0 0.67232 0.6724 0.6726 0.6728 0.6733 0.6745 0.6762 0.6784 50.0 0.65009 0.6510 0.6512 0.6513 0.6516 0.6524 0.6534 0.6546 75.0 0.61397 0.6141 0.6143 0.6145 0.6147 0.6148 0.6148 0.6147
100.0 0.58870 0.5889 0.5894 0.5900 0.5903 0.5901 0.5895 0.5885
"L. Monchick and E. A. Mason. J. Chern. Phys. 35, 1676 (1961); table used with permission from Re1d and Sherwood.121
'T*=kT/a.li=(dipole moment) 2/2a0a 3
'The values offl, m this column differ slightly from values in Table B.1 at low values ofT*.
Appendix B
Notation for the Tables
T Temperature (K) T* kT/e V Molar volume (m 3 kmol- 1 )
b Dipole moment parameter e/k Potential parameter (K) O" Collision diameter (A)
References for the Tables
309
n Collision integral
Subscripts c critical
D diffusion fl viscosity
I. R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York (1960).
2. R. C. Reid and T. K. Sherwood, The Properties of Gases and Liquids, 2nd edition, McGrawHill Book Co., New York (1966).
3. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954).
4. L.A. Bromley and C. R. Wilke, Ind. Eng. Chern. 43, 1641 (1951).
Appendix C
Derivation o.f Proposed Four-Flux Radiation Model*
In the following analysis, it was assumed, for the sake of simplicity, that the gas medium is totally transparent, so that the absorption coefficient and scattering coefficient are solvely functions of the number of particles in a unit volume of the gas-particle cloud. The contribution of absorption by gaseous components of the medium, though small compared to the contribution by particles, can be added later as a correction.
In order to formulate a model that represents the actual conditions that exist in a pulverized-coal flame, namely, the presence of "large" particles that scatter radiation, an effort is made to treat scattering as anisotropic. This is achieved by introducing the forward-, backward-, and sidewise-scattering components, which represent the fraction of scattered radiation in each of the corresponding directions.
The forward-scattering component is defined as
f+1tj2
f = n P(e) sine de co·s2 e -Tt/2
f1t/2
= 2n 0
P(e) sin e cos2 e de (1)
where the notation is that of Chapter 5. The backward-scattering component is defined as
b =n f-"12 P(e) sine de cos2 e 1t/2
*Sneh Anjali Varma, Graduate Research Assistant, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah
311
312
= 2n f" P((}) cos2 (} sin (} d(} 1t/2
and the sidewise-scattering component is
s=(l-f-b)/4
Appendix C
(2)
(3)
For the isotropic case, the above fractions degenerate to f= b= s=f>. For detailed discussion, see reference 6 of Chapter 5.
The phase function, P(8), is a function of particle size, index of refraction, and wavelength of radiation. For spherical particles, the angular distribution is symmetrical about the direction of the incident radiation.
To begin with, a six-flux model is based on drawing energy balances for six discrete components of intensity of radiation in six orthogonal directions, and then, by invoking the condition of axial symmetry, reducing the equations to a four-flux model. In order to accommodate the geometry of the combustor of interest, a cylindrical-polar coordinate system is employed.
Consider a small volume element dV located at a point P(R, 0), as shown in Figure 1. The intensity of I(R, 0) is represented by six discrete components in the direction of, and opposite to the direction of, the three major directions, which are axial, radial, and angular.
Now, consider the energy balance for the intensity vector I:, directed in the positive z-direction. Change in intensity during passage through the small volume is given as
U: +di;- /dz)-I;- =di;- /dz (4)
Loss in intensity due to absorption by the matter in the small volume is
(5)
and loss in intensity due to scattering by the matter in the small volume is
(6)
The fraction of scattered radiation in the direction of the forwarddirected intensity vector is f K.I: and the rest is scattered in other directions, and thus considered lost. The net lost scattered radiation is then
-(1-.f)K.J: (7)
Similarly, there will be an addition to the forward-directed intensity by the backward-scattering component of intensity vector I;, moving in the opposite direction. This increase is
+bK.I; (8)
The intensities traveling perpendicular to the z-direction will contribute a fraction of their out-scattered radiation to the positive z-direction, given
Appendix C 313
HR • .fi.)
a
Ii
----------b
Figure 1. Volume element for radiative analysis. (a) The angular distribution of intensity. (b) The distribution of intensity in six discrete components.
in terms of s, as
(9)
There will also be a contribution to intensity in the positive z-direction by the radiation emitted by matter in the small volume. Considering this emitted radiation to be uniformly distributed in all directions, it may be represented by six equal, discrete components in the six orthogonal directions. Thus, the contribution of emitted radiation in the positive z-direction is
(10)
Summing up all the above terms, an energy balance for radiative transport in the positive z-direction is given as
dl: /dz= -Kai: -(1-f)K.I: +bK.I; +SK.(I: +I; +Iii +Ii) +(Ka/6)Ib(T) (11)
314 Appendix C
or, combining terms:
di; /dz= [Ka+(1-f)K.]I; +bKJ; +SK.(I: +I; +It +Ii)+Kj6[Ib(T)] (12)
Replacing (Ka + K.) by K 1 and K./ K 1 by W0 , the above equation can be rewritten as
(1/Kr)(di: /dz)= -(1- WoJ)I: + Wobi; + WoSUr+ +I;+ It+ Ii) +(1/6)(1- W0 )h(1) (13)
By writing similar energy balance equations in the other directions, the following equations can be obtained for each direction:
-(1/Kr)(di; /dz)= -(1-Wof)I; + W0 bi; + W0 S(I: +I;+ It+ Ii)
+(1/6)(1- W0 )Ib(T) (14)
(1/Kr)[d(I: 0 r)/dr]= r[ -1(1- W0 f)I: + W0 bir- + W0 S(I; +I;+ It+ Ii)
+ (1/6)(1- W0 )I b(T)] (15)
-(1/Kr)[d(I; 0 r)/dr] =r[ -(1- W0 f)I; + W0bi: + W0 S(I; +I;+ It+ Ii)
+(1/6)(1- W0 )Ib(T)] (16)
(1/rKr)(dit /dfJ)= -(1- Wo.f)It + W0 bi0 + W0 S(I; +I;+ I:+ I;)
+ (1/6)(1- W0 )I b(T) (17)
-(1/rKr)(dii /dfJ)= -(1- W0f)Ii + W0bit + W0 S(I; +I;+ I:+ I;)
+ (1/6)(1- W0 )I b(T) (18)
The assumption of axial symmetry results in the following conditions:
(19)
and (20)
By applying these two conditions in the last two equations of the sixflux model, there results
It =Ii =[W0S/(1- W0 f- W0b}](I: +I; +I: +I;) +(1/6)[(1- W0)/(1- W0 f- W0b)]Ib(T) (21)
Substituting these expressions for Ir+ and I; in one of the remaining flux equations gives
(1/K1)(di: /dz)=I: { -(1- W0 f)+ [2W6S2/(1- W0 f- Wob)]} +I; {W0b+ [2W6S2/(1- W0 f- Wob)]} +(I: +I;){W0 S+ [2W6S2/(1- Wof- Wob))}
Appendix C 315
+ Ib[(1/6)(1- W0 )]{1 + [2W0S/(1- W0f- W0 b)]} (22)
(1/K 1)(di: /dz) = C 1I: + C 2I; + C3(1: +I;)+ C4I b(T) (23)
where
C 1 = -(1- W0 f)+ [2W6S2/(1- W0f- W0 b)] (24)
C2 =W0b+[2W6S2/(1-W0 j-W0b)] (25)
C 3 =W0 S+[2W6S2/(1-W0 f-W0b)] (26)
and
C4 = [(1- W0 )/6]{1 + [2W0S/(1- W0f- W0 b ]} (27)
The complete four-flux model can now be written as
(1/K1)(di: /dz)= C 1I: + C2I; + C3(1: + Ir-)+ C4Ib(T) (28)
-(1/K1)(di; /dz)= C 1I; + C 2I: + C3(1: + Ir-)+ C4 Ib(T) (29)
(1/K1)[d(Ir+ · r)/dr] = r[ C 1I: + C2I; + C 3(J,: +I;)+ C4 Ib(T)] (30)
and
Appendix D
Derivation of Eulerian Finite-Difference Equations*
In this appendix, the general Eulerian finite-difference equation is derived in two-dimensional, steady, cylindrical coordinates. The flow field of interest is subdivided into computational cells by some grid pattern. Figure 1 displays a typical internal main cell where the 4>-equation [Eq. (2), Chapter 14] can be integrated over the volume obtained by rotating the area represented by the dashed lines about the symmetry axis to give r: rn s: {(a;ax)(pur4>)+(8/8rXpvr4>)-(a;ax)[rr(84>/8x)]
-(8/8r)[rr(84>/8r)]-rS<1>} dx dr d~ 3 =0 (1)
where the third coordinate, ~ 3 , has the integration limits of 0 to 1 radians for convenience instead of 0 to 2n radians, because of assumed axial symmetry.
Considering the first convection term in Eq. (1)
fXe f'n e (8/8xXpur4>) dx dr d~3 Xw r8 J 0
and noting that all properties are uniform in the third direction, performing two formal integrations gives
frn [purc/> J: dr rs
where e and w represent the expression to be evaluated at the east and west faces, respectively. As with any finite-difference development, the derivation
*John J. Wormeck, Senior Engineer, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah
317
318
NW •
we
• sw
N •
nw n ne r-------------~
I I I I I
I I I
:e
L ----- - ---- - - - _J sw s se
• 5
Appendix D
NE •
• SE
Figure 1. Illustration of the grid symbols for a computational cell.
is somewhat arbitrary; the following method yields the most accurate results. From the mean-value theorem:
f+t.r f(r) dr~J(f)(ilr) (2)
where r<r<r+ilr, and as r---+0 convergence is assumed. Applying this theorem to the first convection term yields
(puro/).,(rn -r.)-(puro/)w(rn -r.)
where each subscript represents evaluation at that particular face. Grouping the geometric terms gives
(puo/ ).,A., - (puo/ )wAw
with the following definition.s
A.,= r.(rn -r.)
Aw=rw(rn-rs)
(3)
which are the areas of the east and west faces of the cell, respectively, as
Appendix D 319
shown by considering
f rn Jl A,= rdrdC 3 =(r;-r;)/2
'n 0
or
A,= (rn -r.)(rn + r.)/2 = rp(rn -r.)
Furthermore, since re = r w• these areas are equal and only one symbol will be used:
(4)
The numerical procedure TEACH< 1l employs a staggered grid system,<2 l
where the velocities are stored midway between the grid lines; that is, at the exact locations which are required. The first convection term [Eq. (3)] becomes
Convection coefficients are defined as
CE = PeUEA.:w
Cw = PwUpA,w
which gives the mass flux through the face corresponding with the subscript. Both p and 4> are defined at the main grid nodes and some sort of inter
polation is needed to determine their values at the faces midway between node points. The practice with TEACH is to linearly interpolate dependent variables and use simple averaging for fluid properties; thus
Pe = ( PP + PE)/2,
and
P<¢<E: ¢=(1-fE)4Jp+fE¢E, fE=(X-Xp)j(XE -Xp)
W<¢<P: 4>=0-fw)¢w+fw¢P, fw=(x -xw)/(xp-Xw) P<¢<N: 4> = (1-fN)4Jp + fNcf>N, !N= (r -rp)j(rN -rp) S <4> <P: ¢=(1-fs)4>s +fs¢p, fs =(r-rs)/(rp-rs)
Using these relationships, the convection coefficients are
CE = (pp + PE)UEAew/2
and the first convection term becomes, upon substitution:
(5)
(6)
(7)
(8)
320
Similarly, the second convection term in Eq. (1) is
fxefrn f1 (o/or)(pvr¢) dx dr d~ 3 Xw rs J 0
Again, two integrations can be performed formally to give
r: [pvr¢ ]~ dx
and from the mean value theorem
(pvc/J)nrn(Xe -xw) -(pvc/J),r,(Xe -Xw)
where the geometric terms are
An= rn(Xe- Xw) A,=r,(Xe-Xw)
Appendix D
(9)
and in this case are different, and hence the convection coefficients are defined as
CN = PnVnAn = (PN + pp)VNAn/2 Cs = p,v,A, = (Ps + pp)VpA,/2
(10)
Therefore the final form of a second convection term of the ¢-equation becomes
CNfN¢N-Cs(1-fs)¢s + [CN(l-fN)-Csfs]cPP
Considering the diffusion terms in Eq. (1) separately
(11)
r: rn f { -(8/ox)[rr(o¢/ox)] -(o/or)[rr(o¢jor)]} dx dr d~3 (12)
and integrating twice gives
rn - [rr(o¢jox)]~ dr-r: [rr(o¢/or)]~ dx
Using the same technique as presented for the convection terms, these last integrals can be evaluated as
-re(o¢/ox)ere(rn -r,)+r w(o¢jox)wrw(rn -r,)
- rn(oc/Jjor)nrn(Xe- Xw) + r,(oc/Jjor),r8(Xe- Xw)
As expected, the same geometric quantities appear as the convection terms; substituting Eqs. (4) and (9) yields
- re(o¢/ox)eAew + r w(o¢/ox)wAew- rn(o¢/or)nAn + r.(o¢/or).A.
The derivatives at the four faces must be expressed in terms of variables at
Appendix D 321
main node points. Employing central differences (which are second-order accurate)<3> gives
- r e[(cj>E -cj>p)j(DXpE)]Aew + r w[(cj>p -c/>w)/(DXpw)]Aew
- rn[(cf>N -cj>p)/(D.YNp)]An + r.[(cj>p -c/>s)/(byps)]A.
where the bx and by stand for coordinate distance between the node points indicated by their corresponding subscripts.
Diffusion coefficients can be defined as
DE~ re(Aew)/(DXpE)=(r p+ r E)Aew/(2 DXpE)
Dw= r w(Aew)/(DXpw)=(rp+ r w)Aew/(2 DXpw)
DN = rn(An)/(D}'Np) = (r p + r N)An/(2 D}'Np)
Ds = r.(A.)/(Dyp5 ) =(r p+ r s)A./(2 Dyps)
Thus the diffusion terms can be expressed as
-DE(cJ>E -cj>p) + Dw(cJ>p -c/Jw) -~(cf>N -cj>p)+ Ds(cJ>p-cJ>s) (13)
where the similarity with the control volume formulation is noted. The exchange coefficients and geometric quantities are contained in the diffusion coefficients D, while the difference in 4> which drives the diffusion is explicitly shown with the correct sign, such that 4> enters the cell when the ¢-difference is negative.
Finally, considering the source terms in Eq. (1):
fxe frn il rS<I> dx dr d~ 3
xw r5 0
One of the major techniques responsible for success of the TEACH formulation is to express this source term as linear in the dependent variable. Thus
fxe frn il rS<I> dx dr d~ 3 =St+SicJ>P
Xw rs 0 (14)
which defines two source term coefficients, st and Si. If the source term happens to be nonlinear in terms of the dependent variable, cj>p, the technique calls for just cj>p to be factored from the expression (if possible) and to appear with the st coefficient in Eq. (14); i.e., cj>p appears implicitly, while the remaining factored expression involving cj>p will be considered as known (explicit-based on old values) and lumped together in the st coefficient of Eq. (14). Furthermore, as shown in the stability analysis,<2 > st must be negative to guarantee stable convergence.
In general, s<~> will be functions of all the dependent variables and other fluid properties as well as various types of derivatives involving these quantities. When integrating this source term, the value prevailing at the cell center will be used for all quantities and any derivatives will be evaluated
322 Appendix D
by central differencing. Therefore, the source term is considered constant, giving
(15)
where .1 V is the volume of the cell. Upon substitution of these newly defined coefficients [Eqs. (8),(11 ),
(13), and (15)] into Eq. (1) the general ¢-equation becomes
[CE/E -DE]cf>E -[Cw(1-fw)+ Dw]cf>w+ [CN/N -~]cf>N-[Cs(1-fs)+ Ds]cf>s
+ [ CE(l-/E)+ DE- Cwfw+ Dw+ CN(1 -/N) + ~-Csfs + Ds]cf>P
=Su+Spcf>p
Adding and subtracting CE-Cw+CN-Cs from the expression in brackets preceding cf>p and rearranging to obtain common expressions gives
[(CE-Cw+CN-Cs)+Dw+(1-fw)Cw+DE -fECE
+Ds +(1-fs)Cs +~-fNCN]cf>p
= [Dw+(1-fw)Cw]cf>w+ [DE -fECE]cf>E + [Ds +(1-fs)Cs]cf>s
+ [DN-fNCN]cf>N+Su+ Spcpp
It is convenient to define new total coefficients to replace these common expressions:
AE=DE-/ECE
Aw=Dw+(1-fw)Cw
AN=~-fNCN
As =Ds -(1-fs)Cs
(16)
In terms of these total coefficients, the finite-difference form of the ¢-equation becomes
[CE -Cw+ CN-Cs +AE + Aw+ AN+ As]cf>p=AEcf>E + Awcf>w+ AN<f>N +Asc/>s+Su+Spcpp (17)
Numerical stability considerations lead to two further modifications of the above equation, which finally reduces to
[ AE + Aw+ AN+ As]cf>p= AEcf>E + Awcf>w+ AN<f>N + Asc/>s + Su+ Spcpp (18)
The above analysis is for illustrative purposes only; the interested reader is referred to a lengthy but exhaustive treatment< 2> of the procedure including the formulation of both velocity and pressure equations, boundary conditions, solution of finite-difference equations, underrelaxation, grid definition, and extensions to three-dimensional time-dependent, general orthogonal coordinates systems.
Appendix D 323
References l. A.D. Gosman and W.M. Pun, Lecture notes for course entitled "Calculation of Recirculating
Flows," Imperial College of Science and Technology, London, Report No. HTS/74/2 (1974). 2. J.J. Wormeck, Computer Modeling of Turbulent Combustion in a Longwell Jet-Stirred
Reactor, Ph.D. Dissertation, Washington State University, Pullman, Washington (1976). 3. P.J. Roache, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New
Mexico (1972).
Index
Absorption coefficient definition, 87 for gas, 100 for gas-particle cloud, 100 for monodisperse particles, 95 overall, 101 for p·_,lydisperse particles, 101 relation to absorption index, 92
Absorption efficiency definition, 93, 99 vs. size parameter of char, 94
Absorption index, 92 Acceleration modulus, 114 Activation temperature, 66 Advection of turbulent kinetic energy,
61 Alkali metals
calcium CaC03 , 196, 198 CaS04, 196, 198 gypsum, 196 reactions of, 197-198 sulfur retention by, 198, 204
sodium NaCl, evaporation and condensation
of, 201 NaOH, 201 Na2 S04, formation and condensation
of, 201 reactions of, 201, 204 sulfur retention by, 198, 204
Ammonia in coal combustion, 186 from devolatilization, 190 gas-phase kinetics, role in fuel-NO,
192-194 gasification, 208 in industrial coking, 188
325
Ammonia (cont.) from pyrolysis of model nitrogen
compounds, 192 in stirred-reactor combustion, 194-195
Anisotropic scattering, 89, 97, 99 Anisotropy, 60,62 Ash
composition, 129 definition, 133 effect on overall absorption coefficient,
101 effect on overall scattering coefficient,
101 effect on radiative transport, 100 equations, 228-229 model assumptions, 220 phase function, 100 shape, 92 trace elements, 130 as a tracer, 13 9
Attenuation: see also Extinction, Interception
defmition, 87, 92, 93 efficiency, 93 efficiency vs. size parameter of char, 94
Benzonitrile, 192 Blowing parameter
heat transfer, 225, 226, 282 mass transfer, 226, 282
Bouger-Lambert's law, 87 Boundary conditions, 102, 241, 268 Boussinesq assumption, 59 Buoyancy, 60
Calcium: see Alkali metals Calcium sulfate: see Alkali metals
326
Carbon monoxide, 183 Carbon-to-hydrogen ratio, 179 Char
appearance, 140 composition,128,137, 220 definition, 131 density, 228 effect on overall absorption coefficient,
101 effect on overall scattering coefficient,
101 enthalpy, 229 formation, 221, 223 index of refraction, 92 oxidation, 217-221, 223 phase function, 100 product enthalpy, 230 radiation coefficient vs. size parameter,
94 SEM photographs, 141 shape, 92 size distribution, 131 size parameter, 100
Char-carbon dioxide reaction activation energy, 157-159 Arrhenius factors, 157-158 controlling resistances, 160 mechanism, 156 "poisoning" due to carbon monoxide,
156 rate expressions
exponential, 158 Langmuir, 156
reaction order, 156 recommendation, 160
Char-hydrogen reaction, 164 Char-nitrogen: see Nitrogen-in-coal Char-oxygen reaction
activation energy, 155 Arrhenius factors, 155 controlling resistances, 152 internal surface area effects, 154 mechanism, 153 rate constants
chemical, 153-155 diffusional, 153
reaction order, 153, 155 recommendation,154
Char-steam reaction activation energy, 161-163 Arrhenius factors, 162-163 controlling resistances, 16 3
Char-steam reaction (cont.) mechanism, 161 "poisoning," 160 rate expressions
exponential, 161 Langmuir, 160
reaction order, 161 recommendation, 163 water-gas shift reaction, 163
Chemical equilibrium, 3-13 Chemical potential, ideal gas, 4 Chlorine
condensation of, 201 evaporation of, 201 HQ, 201 reaction of, 201
aosure problem, 59 Qosure schemes
mean Reynolds stress, 62 mean turbulent energy, 61-62 mean velocity field, 59-61
CN, kinetic role in fuel-NO, 192-194 Coal
ASTMrank anthracite, 124,126, 129 bituminous, 124, 126, 129 lignite, 123-124, 126, 129 meta-anthracite, 124 subbituminous, 124, 126, 129
characteristics, 123 classification, 123-125 composition, 123 formation, 123 heating value, 124 properties
density, 228 enthalpy, 229
reaction kinetics, 221-224 model assumptions, 220 modeling, 217-232 observations regarding, 218-219
SEM photograph, 141 size distribution, 131 specific gravity, 126 specific heat, 126 structure, 131-132 swelling index, 127 thermal conductivity, 126
Index
Coal-·dust flames: see Propagating flames Coal-mine explosions, 235-236, 248 Coal particle: see Particle
Index
Coalescence/dispersion mixing model, 7 5 Collision functions, 2 84 Combustion
of methane-air, 172 of volatiles, 169, 179
Condensed phases, 9 Continuity equation, differential
discrete phase, 22-23 gas-phase
in gas-particle mixture, 21-22 mixture of species, 20 species, 23
overall, 23 Continuity equation, macroscopic
gas-phase, in gas-particle mixture, 40-41 overall, 41 particulate phase, 41
Contact index, 68, 73, 75, 250 Convergence criteria, 12, 241, 254, 291 COS,197,202,203,205 Cross sections
for angular scattering, 93, 99 for total scattering, 93, 99
cs2 ,202,205 Cyanogen, 190
Density bulk density
gas, 21 particles, 22
gas mixture, 19 gas-particle mixture, 23 mass density of species i, 19 molar density of species i, 19
Devolatilization, 217, 218, 222, 239, 242, 244,246,247, 252;see also Pyrolysis
Diffuse particles, 100 Diffuse surface, 87, 102; see also Lambert
surface Diffusion
as a rate-limiting process, 245-24 7 transport of enthalpy, 30, 3 7 transport of kinetic energy, 30
Diffusion approximation method, 84, 98 Diffusivity, gas
basis, 46 binary, 51 equations, 51 mixtures, 51, 283 multicomponent, 51-52
Dissipation function, 273 of turbulent kinetic energy, 61, 273
Distribution of particles, 245
327
Drag coefficient, particle, 110, 219;see also Particle drag
Eddy viscosity, 59-62 Efficiency factor
for absorption, 93, 94, 99 for extinction, 93, 94, 99 for scattering, 93, 94, 99
Emission coefficient, 86 Emissivity, 100, 102 Energy density, 86 Energy equation, differential
discrete-phase, 33-38 gas-particle mixture, 31-33 gas-phase
mixture of gaseous species, 28-31 thermal, 31
overall, 38 Energy equation, macroscopic
gas phase, in gas-particle mixture, 43-44 overall, 45 particulate phase, 44-45
Enthalpy ideal gas, 4, 10 zero of, 11
Equations of motion, 58 Equilibrium
chemical, 3-13 multicomponent, 3-13
Equilibrium temperature, 101 Equilibrium theory of turbulence, 61 Equivalence ratio, 242 Eulerian formulation, 15-18 Eulerian reference frame, 266 Explosions: see Coal-mine explosions Extinction, 87; see also Attenuation Extinction coefficient, 94
"f-g" equations, 71 FeS, 197 Film theory, 225 Finite-difference equations, 268 Fixed carbon, 134 Flame thickness, 236,241, 243, 251 Flame velocity: see Propagation velocity Fluorine, 183
328
Flux method basis, 98 four-flux, 98, 99 for furnaces, 84 six-flux, 99 two-flux, 98
Fluctuations, 59-62,67, 75 Fly ash
formation mechanisms cenospheres, 199-200 dense flyash, 198-199 submicron flyash, 200-201
gasification, 208 size distribution, 195
Force aerodynamic, 25, 27, 32, 34,44 body,24,26,27,44 surface, 24
Four-flux radiation model, 285 Fuel-nitrogen: see Nitrogen-in-coal Fuel-NO: see Nitric oxide
"Garbage" estimates, 13 Gas absorption coefficient, 101 Gas emissivity, 101 Gas-phase
density, 226, 227 diffusivity, 226, 227 flames: see Propagating flames heat capacity, 226, 227 mixture properties, 226 oxidation: see Hydrocarbon oxidation thermal conductivity, 226, 227
Gasification entrained-flow, 207 K-T process, 208 pollutantformation,207
Gauss-Seidel iteration, 269 Gibbs function
ideal gas, 4 minimization of, 3-9
Global reactions, 178-179 Gray body, 87 Gfay boundary, 102 Gray medium, 96, 100
Heat transfer conductive, in gas, 30, 32,44 gas and particle, 225 moisture vaporization control, 224
Heat transfer (cont.) radiative, 32, 44,45 on separated surface, 35, 36
Heating rates rapid, 169-171 slow, 169-171
Heavy metals condensation of, 195, 199, 200 list of, 195, 200 toxicity, 183 vaporization of, 199, 200
Index
Heterogeneous reactions of carbon (char) activation energy
apparent, 150 true, 150
with carbon dioxide: see Char-carbon dioxide reaction
controlling resistances bulk phase mass transfer, 150 chemical reaction, 150 pore diffusion, 150
diffusional effects, minimization of, 150 with hydrogen: see Char-hydrogen
reaction impurities, catalytic effect of, 152 model assumptions, 149 with oxygen: see Char-oxygen reaction reaction order, 150 relative rates, 150 with steam: see Char-steam reaction temperature regimes, 150 variables influencing rates, 151
HOCN, 193 Hottel's zone method, 84, 97, 98 HS, 204,205 HS02, 204 H2S04, 205 Hydrocarbon, 239, 245 Hydrocarbon oxidation, 217, 219-221,
224 Hydrogen cyanide
in coal combustion, 186 from devolatilization, 190-191 gas-phase kinetics, role in fuel-NO,
192-194 gasification, 208 from pyrolysis of model nitrogen
compounds, 192 role in prompt NO, 187 in stirred-reactor combustion, 194-195
Hydrogen sulfide decomposition of, 204-205
Index
Hydrogen sulfide (cont.) from devolatilization, 203, 204 gasification, 209 oxidation of, 204-205 from pyrite, 197 reaction with tars, 204
Ignition, 219 Input parameters, 242, 243 In-scattering, 97, 99 Integral scale of turbulence, 62 Intensity
average, 86 definition, 85 homogeneous, 85 integrated, 86 isotropic, 85, 89
Interception, 87; see also Attenuation Intermittency, fuel-air, 7 3 Iron reactions
C02 formation, 200 from pyrite: see Pyrite
Isotropic scattering, 97 Isotropic turbulence, 249 Isotropy, 60,62
Kirchhoff's law, 87 Knudsen number, 111
Lagrange multipliers method of undetermined, 5 nondirnensional, 6
Lagrangian formulation, 15-18 Lagrangian reference frame, 266 Lambert surface, 102; see also Diffuse
surface Laminar flames: see Propagating flames Laminar flow, 57 Lennard-Jones potential, 284
Macromixing, 69 Macroscale of turbulence, 57, 249 Macroscopic equations of change, 253 Mass flux of gaseous species, 19-20,40 Mass transfer
diffusion, 245-247 gas particles, 226
Mean beam length, 101
Mean free path, 46, 60 Mean speed, 4 7 Methane-air flames: see Propagating
flames Methyl cyanide, 192 Micromixing, 69 Microscale of turbulence, 57, 249 Mie coefficient, 93 Mie scattering, 89 Mie theory, 89, 99, 100 Mineral matter
composition of CaC03 , 196 CaS04, 196 clay, 196 elemental, 196 ferric and ferrous sulfates, 196 gypsum, 196 kaolinite, 196 pyrite, 195-196 quartz, 196 total inorganic sulfur, 202
concentration of, 195 size of, 195
Mixing length, 60 Models
multidimensional, 263 one-dimensional, 235 pulverized-coal, 217 pyrolysis, 140 quasi-global, 17 8-179 reaction, 172 theoretical, 16 9
Modified scattering efficiency, 100 Modules, program, 264-266 Moisture
density, 228 enthalpy, 229-230 gas-particle, 226 vaporization, 221, 224
Momentum equation, differential discrete-phase, 26-28 gas-phase
in gas-particle mixture, 25-26 mixture of gaseous species, 23-25
overall, 28
329
Momentum equation, macroscopic gas-phase, in gas-particle mixture, 41-42 overall, 43 particulate phase, 42-43
Monte Carlo method, 84, 97 Multiple scattering, 89
330
Newton-Raphson correction equations, 6-9,76
Newton-Raphson functionals, 6 NHand NH2
in coal, 188 gas-phase kinetics, role in fuel-NO,
192-194 Nitric oxide
exhaust levels, 186 formation from atmospheric N2
flame radicals, effect of, 187 general behavior, 186 from nitrous oxide mechanism,
186-187 prompt NO, 187-188 sulfur, effect of, 206 thermal NO, 187 turbulent fluctuations, effect of, 187 from Zeldovich mechanism, 186-187
formation from coal-nitrogen (fuel-NO) gas-phase kinetics, 192-194 in stirred-reactor combustion,
194-195 Nitrogen
bimolecular from coal-nitrogen, 190, 194
gasification, 207 Nitrogen dioxide, 187-188 Nitrogen-in-coal
concentration of, 188 oxidation of
air mixing, effect of, 191 of char-nitrogen, 190-191 of volatile nitrogen, 190-191
structure of, 188 volatilization of
in industrial coking, 188 by rapid heating, 189-190 by slow heating, 189 by very rapid heating, 191
Nitrogen tars, 190 Nitrous oxide
in atmosphere, 186 from atmospheric N2 , 187 exhaust levels of, 186 from fuel-nitrogen, 194
NOx: see Nitric oxide, Nitrogen dioxide Nusselt number, 116, 282;see also
Particle-gas heat transfer
OCN, 192
1-DICOG model assumptions, 253 auxiliary equations, 253-256 comparisons, with experiment, 257 differential equations, 254 predictions, 254-258 solution technique, 254
One-dimensional models, 235-262 Optical properties of components, 99 Overall absorption coefficient, 101 Oxidation
of acetylene, 176-177 of carbon monoxide, 178 of ethane, 176, 178
Index
of ethylene, 176, 178 high-molecular-weight hydrocarbons,
177 hydrocarbons, 169, 176 of hydrogen, 170 of methane, 170-177 of propane, 177
Particle absorption coefficients, 94 ash, 92, 97, 99,100, 103;seealso Ash char, 92, 97, 99, 100, 103; see also Char diameter, 217, 218, 220 efficiency factor, 93 gray, 96,100 large, 97, 98 mass, 228 nonspherical, 92 phase function, 100 products enthalpy, 229 properties
density, 228 enthalpy, 229 heat capacity, 228
refractive index, 92 scattering coefficient, 94 shape, 92 size, 95, 96, 99 size parameter, 91 soot, 91, 95 swelling, 218, 220
Particle diffusion, 107-110 Particle drag
Basset force, 114 buoyancy, 113 steady-state aerodynamic drag
Mach number effects, 110
Index
Particle drag (cont.)
mass transfer effects, 111 particle asphericity effects, 112 rarefaction effects, 111 Reynolds number effects, 111
virtual mass, 113 Particle-gas heat transfer
convective Mach number effect, 116 mass transfer effect, 116 Reynolds number effect, 116 rotation effects, 117
radiative heat transfer, 117 Particle-gas mass transfer, 11 7 Particle lag, 239, 253 Particle lift forces
Magnus effect, 114 Saffman lift, 115
Particle scattering, 89-94 Particle sphericity, 113 Perfectly stirred reactor, 70, 289 Phase function
for char and ash particles, 100 definition, 95 in radiative transport equation, 95
Pollutants combustion, 184-208 gasification
inorganics, 208 nitrogen, 207 particulate, 208 sulfur, 209 trace elements, 208
mechanisms, 170 Polydispersed: see Distribution of
particles Polynuclear aromatics, 184 Prandtl hypothesis, 60 Prandtl-Kolmogorov relation, 61, 272 Prandtl number, 116, 289 Prandtl-Schmidt numbers, 62 Predictor-corrector, 254 Premixed flames: see Propagating flames Pressure
partial, 29 stress, 24
Probability density function, 70, 275, 290 Production of turbulent kinetic energy, 61 Propagating flames
laminar, 235-250 mechanism, 248-251
Propagating flames (cont.) model
assumptions, 237 comparisons, with experiment,
242-245 equations, 238-240 solution, 240-241
turbulent, 248-251 Propagation mechanism, 248-251 Propagation velocity
331
laminar, 236, 241, 244-245 turbulent, 250
Proximate analysis, 126 -128 Pyridine
in coal, 188 combustion of, 194-195 from devolatilization, 192 pyrolysis of, 192
Pyrite: see also Mineral matter oxidation of, 197 reaction with CaC03 , 197-198 reductive decomposition of, 197, 203 thermal decomposition of, 197
Pyrolysis: see also Devolatilization data, 135-140 experiments, 134 fast, 133 kinetics, 143 methods, 134 models
kinetic, 142 series, 143 single-step, 14 2 two-step, 142-143
observations, summary of, 144 products, 139-140 rate of, 135 two-step, 135
Pyrolysis products: see Pyrolyzate composition
Pyrolytic graphite, 185 Pyrolyzate composition, 169-171, 176 Pyrolyzates: see Volatiles Pyrrole
in coal, 188 from devolatilization, 192 pyrolysis of, 192
Quasi-laminar flow, 63 Quinoline, 192
332
Radiation basis equation, 84, 95 definition, 85, 95 effects of, 245 energy density, 86 flux,86 heat transfer, 93-106 intensity, 85 modeling,239,242,244,252
Radiation coefficients vs. size parameter for char, 94
Rate global reaction, 178-179 kinetic data, 170, 172-175, 177-178 limiting step, 178
Rate-resolution reactor combustor, 256-257, 267 gasifier, 256, 257-258, 267
Rayleigh scattering, 90 Reaction mechanisms: see also Oxidation
global, 169, 177 overall, 180
Reaction rate gaseous species, 20 particles, 22, 40
Reflectivity, 1 02 Refractive index, 92, 99, 100 Reynolds decomposition, 58 Reynolds equations, 59 Reynolds number
particle, 279 relative, 111, 114, 116, 117 turbulent, 62, 249
Reynolds stress, 59 Reynolds transport theorem, 15-18
sand s2, 205 "S" coordinates: see Streamline Scattering
coefficient, 87, 88, 100 definition, 89, 90 efficiency, 103
Schmidt number, 108, 282 Segregation, scale of, 6 9 Shear stresses, 57, 59-61 Sherwood number, 11 7; see also Particle-
gas mass transfer SiO, 200 Six-flux radiation model, 286 so, 204,205
Sodium: see Alkali metals Soot
formation in flames, 184-185 oxidation of, 185 properties of, 184
Index
SOx: see Sulfur dioxide, Sulfur trioxide Specific heat capacity
ideal gas, 10, 227 mixtures, 227
Stockmayer potential, 285 Streamline,238-240 Stress
"apparent," due to diffusion, 24, 25, 27, 28
on particle due to local strain rate in fluid, 26
on species, i, 29 Sulfur dioxide
exhaust level, 204 formation
gas-phase kinetics, 204-206 oxidation of pyrite: see Pyrite reaction with alkalis, 198, 204 reaction with NOx, 206
Sulfur gasification, 209 Sulfur, gas-phase reactions of, 204-206 Sulfur-in-coal
inorganic CaS04 , 196 ferric and ferrous sulfates, 196 gypsum, 196 pyrite, 195-196 total concentration, 202
organic disulfides, 202 mercaptans, 202 sulfides, 202 thiophene structures, 202 total concentration, 202
volatilization of in industrial coking, 202-203 inorganic decomposition, 202-203 organic, 202-203 pyrite decomposition: see Pyrite residual char-sulfur, 203
Sulfur trioxide behavior, 205 concentration, 206 gas-phase kinetics, 206
Suppressant, 248 Surface tension, 34, 35,44 Swelling: see Particle, swelling
Index
Thermal conductivity, gas basis, 46 equations, 49-50 mixtures, 50, 227 single species, 50, 227
Thermal radiation, 85 wavelength, 99
Trace elements ash, 129-130 combustion, 196 gasification, 208
Trajectories, particle, 279 Transmissivity, 102 Transport coefficients, 45 Transport properties, 283 Tridiagonal matrix algorithm, 269, 287 Turbulence coefficients, 62-63 Turbulent flames, 248-251 Turbulent intensity, 249 Turbulent kinetic energy, 61, 272 Turbulent macroscale, 57 Turbulent microscale, 57 Turbulent Reynolds number, 62 Two-phase flow, 278
Ultimate analysis, 123, 127-128 Unburned hydrocarbons
as pollutants, 183 reaction with NO, 194
Underrelaxation parameters, 12
Vaporization: see Moisture, vaporization
Velocity diffusion, 19, 24 mass-averaged
gas, 19 particles, 22, 27
mixture, 23 mole-averaged, 19
Vinyl cyanide, 192 Viscosity
basis, 46 eddy,59-62 equations, 47-48, 227 mixtures, 48-49, 227 molecular, 57 turbulent effective, 59-62
Volatile nitrogen, 190-191 Volatiles
combustion, 169, 179 composition,137, 169-171, 176 defmition, 133 product enthalpy, 230
Work rate due to aerodynamic forces, 34 due to body forces, 29 due to particle dilatation, 34 due to surface forces, 29 flow, 34,38
333
Zone method, 84, 97, 98; see also Hottel's zone method